## Curvature of an Ellipse

The curvature of a curve
$y=f(x)$
is given by
$\kappa = \frac{\frac{d^2y}{dx^2}}{(1+ (\frac{dy}{dx})^2)^{\frac{3}{2}}}$
.
An ellipse can be written parametrically as
$(acos \theta , b sin \theta )$
.
$\frac{dy}{dx} = \frac{dy/ d \theta}{dx/ d \theta}= \frac{b cos \theta}{-a sin \theta}= \frac{-b}{a} cot \theta$
.
Finding
$\frac{d^2 y}{dx^2}$
is a little trickier. We use
$\frac{d}{dx} = \frac{d \theta}{dx} \frac{d}{\theta}= \frac{\frac{d}{\theta}}{\frac{dx}{d \theta}}= - \frac{1}{asin} \frac{d}{d \theta} = - \frac{1}{a}cosec \theta \frac{d}{d \theta}$
.
Then
$\frac{d}{dx} (\frac{dy}{dx})= - \frac{1}{a}cosec \theta \frac{d}{d \theta} ( \frac{b}{a}cot \theta)= \frac{b}{a^2}cosec^3 \theta$

Finally
$\kappa = \frac{\frac{b}{a^2}cosec^3 \theta}{(1+ (\frac{-b}{a} cot \theta)^2)^{\frac{3}{2}}}$